Game Theory, Continued: From Zero-Sum to Non-Zero-Sum. Problem Set 3 due on FRIDAY!

Transcription

1 Game Theory, Continued: From Zero-Sum to Non-Zero-Sum Problem Set 3 due on FRIDAY!

2 Blue Cooperate Red Defect Cooperate Defect

3 Game Theory: Basic Taxonomy Zero- vs. non-zero sum Two- vs. N-person games Finite vs. infinite number of choices Iterated vs. non-iterated games Games of perfect information

4 Two-Person Zero Sum Games The notion of a dominant choice The solution of a zero-sum game The value of a game The minimax theorem Pure vs. Mixed strategies

5 Blue A B A 5 3 Red B 4 1

6 Blue Worst I can do Red Worst I can do

7 Scissors Paper Stone Scissors Paper Stone 1-1 0

8 The Minimax Theorem in Game Theory Applies to zero-sum, two-person finite games. The minimax theorem says that in such a game, there is a value V for the game (the same value V for both players). Given an optimal strategy (possibly a mixed strategy), each player can be assured (on average) of obtaining at least V for the game regardless of what the other player does. What this means, essentially, is that both players can examine such a matrix and determine beforehand (and regardless of the other player s plan) what they need to do to ensure receiving an average of V for the game.

9 A Non-Zero-Sum Game Blue Red

10 Blue Red

11 Clyde Clam up Bonnie Stool Pigeon Clam up 1 year 1 year 0 years 20 years Stool Pigeon 20 years 0 years 10 years 10 years

12 Blue Cooperate Red Defect Cooperate Defect

13 Blue Red

14 Blue Red

15 Blue Red

16 The Ultimatum Game : adding intangibles to a utility function Two players, Red and Blue. Red is given $1000 and is told that he can choose an amount to be taken out of this total to offer Blue. Blue then chooses whether to accept this amount or not. If Blue rejects Red s offer, both get nothing.

17 Axelrod s idea: hold a computer tournament in which each contestant is a program that will play a series of rounds of the Prisoner s Dilemma game. Researchers can send in a program to enter in the tournament: each program will play all the others in a series of about 100 successive rounds of the PD game.

18 Some Simple Axelrod- Tournament-like strategies All-Defect simply defects on every round Poor-Trusting-Fool simply cooperates on every round Random is a test strategy that simply cooperates or defects randomly Unforgiving cooperates initially until the first time that the other player defects; after that, Unforgiving defects forever

19 The Tit-for-Tat Strategy Tit-for-Tat cooperates on the first round. Thereafter, on every subsequent round, it simply imitates what the other player did on the previous round. (That is, Tit-for-Tat does at round N what the other player did at round N-1.)

20 A British staff office on a tour of the trenches remarked that he was astonished to observe German soldiers walking about within rifle range behind their own line. Our men appeared to take no notice. I privately made up my mind to do away with that sort of thing when we took over; such things should not be allowed. These people evidently did not know there was a war on. Both sides apparently believed in the policy of live and let live. Dugdale 1932, quoted in Axelrod, p. 74

21 The high commands of the two sides did not share the view of the common soldier who said: The real reason for the quietness of some sections of the line was that neither side had any intention of advancing in that particular district... If the British shelled the Germans, the Germans replied, and the damage was equal: if the Germans bombed an advanced piece of trench and killed five Englishmen, an answering fusillade killed five Germans. Belton Cobb 1916, quoted in Axelrod p. 76

22 The ethics that developed are illustrated in this incident, related by a British officer recalling his experience while facing a Saxon unit of the German Army. I was having tea with A Company when we heard a lot of shouting and went out to investigate. We found our men and the Germans standing on their respective parapets. Suddenly a salvo arrived but did no damage. Naturally both sides got down and our men started swearing at the Germans, when all at once a brave German got on to his parapet and shouted out We are very sorry about that; we hope no one was hurt. It is not our fault, it is that damned Prussian artillery. Rutter 1934, quoted in Axelrod, p. 85